Venn Diagram Word Problems

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Venn Diagram Problems can be challenging for beginners.

It is extremely important to:

Read the question carefully and note down all key
information, usually summarised in dot point form.

Know the standard parts of a Venn Diagram

Work in a step by step manner

Check at the end that all the numbers add up properly.

"A class of 28 students were surveyed and asked if they
ever had cats or dogs for pets at home.

• 8 students said they had only ever had a dog.

• 6 students said they had only ever had a cat.

• 10 students said they had a cat and a dog.

• 4 students said they had never had a cat or a dog.“

Since this question is about cats and dogs, it will require
a two circle Venn Diagram. (See Next Slide)

Our problem is an easy one, where we have been given all
of the numbers for the items required on the diagram.

We do not need to work out any missing values.

All we need to do is place the numbers from the word
problem onto the standard Venn Diagram and we are done.

Eg. Cats Only = 6, Dogs Only = 8, Both Cats & Dogs = 10,
and Neither Cat or Dog = 4

(See Next Slide for completed diagram)

The answer for this question will actually be the same as
the Cats and Dogs in Question 1. However this time we are
given less information, and have to do some working out.

"A class of 28 students were surveyed and asked if they
ever had dogs or cats for pets at home.

• 18 students said they had a dog.

• 16 students said they had a cat.

• 4 students said they had never had a dog or a cat."

Word Problem Two does not contain the word "only"
anywhere in it, and this is an indication that we will need
to do some working out.

The question states that: "18 students said they had a dog“
(without the word "only" anywhere in there).

This means that the total of the Dogs circle is 18.

The total 18 students for Dogs includes people that
have both a cat and a dog, as well as people who only
have a dog.

Some people will not read this question carefully, and will
incorrectly take the supplied numbers and put them straight
into a Venn Diagram like this. (See Next Slide)

Always check at the end that the numbers add up to the "E"
Grand Total. Eg. 16 + 18 + 4 = 38 is much bigger than the
"E" everything total which is 28 students.

This “E” Total is smaller because some students have both
a cat and a dog. We have not accounted for this yet.

Other people might think that we do not have enough
information given to us, and therefore it is impossible to do
this problem.

Let's put down on our diagram all of the information that we
know. (See Next Slide)

All we have left to work out is the number of Cats and Dogs
for the “Intersection” at the centre of the diagram.

We can do this any one of three ways:

Cats and Dogs = Total Cats - Only Cats
OR
Cats and Dogs = Total Dogs - Only Dogs
OR
Cats and Dogs = E Total - Only Cats - Only Dogs - (No cats
and No Dogs)

Any way that we work it out, the answer is 10.

So we now have our Diagram completed. (See Next Slide)

Work out What Information is given, and what needs to be
calculated.

Circles Total = E everything - (No Cats and No Dogs)

Cats Only = Circles Total - Total Dogs

Dogs Only = Circles Total - Total Cats

Cats and Dogs = Cats Total - Cats Only

Finally, check that all the numbers in the diagram add up to
equal the "E" everything total.

"Fifty people were surveyed and only 20 people said that they
regularly eat Healthy Foods like Fruit and Vegetables.

Of these 20 healthy eaters, 12 said they ate Vegetables every day."

Draw a Venn Diagram to represent these results."

This problem is quite different to our other two circle diagrams.

Cats and Dogs are quite different, and needed two separate circles.

However Healthy Foods and Vegetables are not different to each other
because Vegetables are a type of Healthy Food.

We say that vegetables are a "Subset" of Healthy Foods.

This means that we do not separate the circles, and we draw our circles
inside each other like this. (See Next Slide)

"Draw a Venn Diagram which divides the twelve months of the year
into the following two groups:

• Months whose name begins with the letter "J“,
• Months whose name ends in "ber" .

You will need a two circle Venn Diagram for your answer.”

Months starting with J = { January, June, July }

Months ending in "ber" = { September, October, November, December }

The two sets do not have any items in common, and so we will not
need to overlap them. They are “Mutually Exclusive” or “Disjoint”.

The remaining months will need to go outside of our two circles.

There should be all twelve months in the diagram when we are finished.

The working out steps for harder problems are:

Work out What Information is given, and what needs to be calculated.

Check to see if the two sets are "Subsets" or "Disjoint" sets.

If they are "Intersecting Sets" then some of the following
calculations may be required:

Circles Total = E everything - (Not in A and Not in B)
In A Only = Both Circles Total - Total in B
In A Only = The A Circle Total - Total in the intersection (A and B)
In B Only = Both Circles Total - Total in A
In B Only = The B Circle Total - Total in the intersection (A and B)
In the Intersection (A and B) = Total in B - In B Only
In the Intersection (A and B) = Total in A - In A Only

Finally, always check that the numbers in the diagram
all add up to equal the "E" everything total.

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Venn Diagram Word Problems

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